A bounded operator approach to Tomita-Takesaki theory

of the paper we show that the operators Δu and J can, in fact, be associated with any fairly general real subspace of a complex Hubert space, and that many of their properties, for example the characterization of Au in terms of the K.M.S. condition, can be derived in this less complicated setting. In the second half of the paper we show, by using some of the ideas from the first half, that a simplified proof of the Tomita-Takesaki theory given recently by the second author can be reformulated entirely in terms of bounded operators, thus further simplifying it by, among other things, eliminating all considerations involving domains of unbounded operators.